A ball is thrown upwards with an initial velocity of 30 m/s. At what height will its kinetic energy be equal to its potential energy?

Initial velocity of the ball, $u = 30 \text{ m/s}$.
Acceleration due to gravity, $g = 10 \text{ m/s}^2$.

We need to find the height $h$ where the kinetic energy (KE) equals the potential energy (PE).

where $m$ is the mass of the ball and $v$ is its velocity at height $h$.

The total mechanical energy at the start (ground level) is equal to the total mechanical energy at height $h$.
Initial total energy ($E_{initial}$):
At $h=0$, $PE_{initial} = 0$.

Total energy at height $h$ ($E_{at\ h}$):

Since $KE_{at\ h} = PE_{at\ h}$ at this height, we can write:

Cancel out the mass $m$ from both sides:

The height at which its kinetic energy will equal its potential energy is $22.5 \text{ m}$.

Comparing this with the given options:
A. 45.0 m
B. 22.5 m
C. 15.0 m
D. 30.0 m

The correct option is B.

The final answer is $\boxed{\text{B. 22.5 m}}$.